NEW REGULARITY OF KOLMOGOROV EQUATION AND APPLICATION ON APPROXIMATION OF SEMI-LINEAR SPDES WITH H¨OLDER CONTINUOUS DRIFTS

. In this paper, some new results on the the regularity of Kolmogorov equations associated to the inﬁnite dimensional OU-process are ob- tained. As an application, the average L 2 -error on [0 ,T ] of exponential integrator scheme for a range of semi-linear stochastic partial diﬀerential equations is derived, where the drift term is assumed to be H¨older continuous with respect to the Sobolev norm (cid:107) · (cid:107) β for some appropriate β > 0. In addition, under a stronger condition on the drift, the strong convergence estimate is obtained, which covers the result of the SDEs with H¨older continuous drift.


1.
Introduction. Recently, the regularity of the Kolmogorov equation with singular coefficients is applied to study the pathwise uniqueness of stochastic (partial) differential equations (S(P)DEs) with singular drifts, which are, for instance, Hölder continuous, Dini continuous or integrable etc. The main idea is to construct Zvonkin's transform ( [33]) which is a homeomorphism map depending on the solution of the Kolmogorov equation to transform the original S(P)DEs to a new one, where the singular drift is killed with the aid of the Kolmogorov equation and the pathwise uniqueness can be obtained. There are many results on this topic, see [4,7,8,9,10,13,20,28,29,30,31,32], and references therein. Encouraged by this idea, some researchers have adopted Zvonkin's transform to study the strong convergence rate of the Euler Maruyama (EM) method for SDEs with singular drift, for instance, [14,24,25,26] and the SDEs with Jumps [17]. However, so far, there are no results on the numerical method for the semi-linear SPDEs with singular drift. The main difficulty lies in the following. After Zvonkin's transform, compared to the exact solution, the SPDE for the numerical solution contains two additional items produced by the temporal discretization, one of which depends on the trace of the second-ordered gradient operator of the solution to the associated Kolmogorov equation, see (46) and (40) below for more details. In R d , the trace of a linear operator can be controlled by the operator norm, while this is generally not true in infinite dimension case, for instance, the identical operator. Thus, in the SPDEs, the previous results in [28] on the regularity of the Kolmogorov equation, i.e. the estimate for the operator norm of the second-ordered gradient operator of the solution is not available. In other words, to obtain the convergence rate of the numerical method for the semi-linear SPDEs with singular drift, some new regularity, i.e. the trace of the second-ordered gradient operator of the solution to the associated Kolmogorov equation is required. The trace strictly depends on the spectrum of A and the regularity of the drift term b, which will be showed in Theorem 2.7 below by the gradient estimate of the semigroup. This is a new result in related fields.
Through out this paper, let (H, ·, · H , · H ) be a real separable Hilbert space. Denote by L (H) (resp. L 2 (H)) the space of all bounded linear operators (resp. Hilbert-Schmidt operators) on H. Let · (resp. · L2 ) stand for the operator norm (resp. the Hilbert-Schmidt norm). Let (W t ) t≥0 be an H-valued cylindrical Wiener process defined on a complete probability space (Ω, F , t e k , where (β (k) ) k≥1 is a sequence of independent real-valued Brownian motions on the probability space (Ω, F , (F t ) t≥0 , P) and (e k ) k≥1 is an orthonormal basis of H. Fix T > 0 and set f T, are all eigenvalues with counting multiplicities. For any r ∈ R, Let H r = {x ∈ H, |(−A) r x| < ∞} equipped the Sobolev norm x r := |(−A) r x|, x ∈ H r . Then (H r , · r ) is a Banach space and H 0 = H.
Convention: The letter c or C with or without subscripts will denote an unimportant constant, whose values may change in different places. Moreover, we use the shorthand notation a b to mean a ≤ c b. If the constant c depends on a parameter p, we shall also write c p and a p b.
The remainder of this paper is organized as follows: In Section 2, we investigate the new regularity of the Kolmogrov equation associated to the OU-process. In Section 4, as an application of Section 2, we give a result on the average L 2 -error on [0, T ] of exponential integrator scheme for a range of semi-linear SPDEs with Hölder continuous drift. Moreover, under a stronger condition on the drift, the strong convergence rate is obtained.
2. New regularity of Kolmogorov equation associated to OU-process. In this section, we consider the Kolmogorov Equation with singular coefficients: is the Markov semigroup associated to the following O-U process: To characterize the regularity of the solution to (1), we need some assumptions: (A1) There exists a constant α ∈ (0, 1) such that and there exist c > 0, ε ∈ (0, 1) and β ≥ 0 such that Remark 2.1. Under (A1), it is well known that (2) has an up to modifications unique mild solution (Z x t ) t≥0 (see, e.g., [6]) with the associated Markov semigroup (P 0 t ) t≥0 . Remark 2.2. The condition (5) means that the continuity of b t has weaker dependence on the higher dimensional components. More precisely, for any i ≥ 1, On the other hand, for some c 0 > 0. Let b 0 t satisfy (7), (5). Before moving on, we introduce the results on the regularity of (1) in [ If moreover (7) holds, then for any λ ≥ λ T , ∇ 2 u λ T,∞ < ∞ and lim λ→∞ ∇ 2 u λ T,∞ = 0.
Thus it is very hard to obtain (17). We will leave the multiplicative noise case in the future research.
The next theorem gives an estimate for the trace of ∇ 2 u λ we explained in the introduction, which plays a crucial role in analyzing error of numerical schemes.

Approximation of semi-linear SPDEs with Hölder continuous drifts.
The numerical approximation of SPDEs has been a very active field of research. Due to the infinite dimensional nature of state space, in order to be able to simulate a numerical approximation on a computer, both temporal discretization and spatial discretization are implemented. The temporal discretization is achieved generally by Euler type approximations, Milstein type approximations, and splitting-up method (see, e.g., [2,5,12,16,21,27]), and the spatial discretization is in general done by finite element, finite difference and spectral Galerkin methods (see, e.g., [19]). In contrast to substantial literature on approximations of semi-linear SPDEs with regular coefficients, the counterpart with irregular terms (e.g., Hölder continuous drifts) is scarce. Whereas, our goal in this section is to make an attempt to discuss strong convergence of an exponential integrator (EI) scheme, coupled with a Galerkin scheme for the spatial discretization (see (35) and (36) below), for a class of semi-linear SPDEs with Hölder continuous drifts. With regard to convergence of EI scheme for SPDEs with smooth drift coefficients, we refer to [18,22,23] for further details, to name a few. Also, there is a number of literature on approximation of SPDEs with non-globally Lipschitz continuous nonlinearities; see, for instance, [15]. Consider the following semi-linear SPDE on H where A, b, W are introduced in Section 2.
Thus, according to [28, Theorem 1.1], under (A1) and (A2), (33) has a unique mild solution, i.e., there exists a unique continuous adapted process (X t ) t≥0 such that P-a.s., For any n ∈ N, let π n : H → H n := span{e 1 , · · · , e n } be the orthogonal projection, A n = π n A, b (n) t = π n b t and W (n) t = π n W t . With the notation above, we consider the following finite-dimensional approximation associated with (33) on which is the Galerkin projection of (33) onto H n . Since A n x = Ax for any x ∈ H n and b n is Hölder continuous in terms of (6), by virtue of [28, Theorem 1.1], (34) has a unique strong solution. Now we define a numerical scheme to approximate X (n) t in time, which is called discrete-time EI scheme: for a stepsize δ ∈ (0, 1) and each integer k ≥ 0, which is also named as Lord-Rougemont scheme (see, e.g., [14, (3.2)]), where ∆W where t δ := t/δ δ with t/δ being the integer part of t/δ. It is easy to see that Y (n),δ kδ =Ȳ (n),δ kδ for any k ≥ 0.
To obtain the convergence rate of the EI scheme, we need the continuity of b t with respect to t, i.e. (A3) For c > 0 and ε ∈ (0, 1) in (5), The main result of this section is stated as follows.
(1) The average L 2 -error on [0, T ] satisfies (2) If in addition, there exists an γ ∈ ( 1 2 , 1] such that (−A) γ b T,∞ < ∞, then one has the strong convergence estimate: (39) Remark 3.2. If 1 − α < ε 2 , then we can take β = 0 and (5) reduces to (7). Otherwise, to ensure ν > 0, we have to take β > 0. See the example in Section 4 for more details. . Moreover, Theorem 3.1 (2) covers the result of the finite dimensional case, see [14]. In fact, when H is finite dimensional, (−A) γ is a bounded linear operator, and the second term on the right side of (39) disappears. In this case, we can take α = 1, β = 0 and then ν = ε 2 . Thus sup Remark 3.4. To obtain the strong convergence estimate, as in the proof of Theorem 3.1 (2), we need to deal with , then this term can be treated as in (57) below. On the other hand, by Theorem 2.5 (3), (−A) γ b T,∞ implies (−A) γ ∇u T,∞ < ∞. For more details, see the proof of the Theorem 3.1. In fact, the above trick is used to prove the pathwise uniqueness of the neutral functional SPDE, see [11], where the condition [11, (H3)] is something like (−A) δ ∇u T,∞ < ∞ for some δ > 0.
Remark 3.5. To avoid complicated computation, in the present setup we work only on the case that the drift is uniformly bounded. Nevertheless, employing the standard cut-off approach (see, e.g., [1]), we of course can extend our framework to the setting that the drift coefficient is unbounded.
The following lemma provides us with a regular representation of the continuoustime EI scheme (36). Lemma 3.6. For any t ∈ [0, T ] and λ ≥ λ T , it holds that in which I is the identity operator on H.
Proof. Since A n x = Ax for any x ∈ H n , (36) can be reformulated as For any λ ≥ λ T , let u λ t be the solution to (1), then by Theorem 2.7, for any x ∈ ∪ ∞ n=1 H n , where ∇ 2 ei := ∇ ei ∇ ei , the second order gradient operator along the direction e i . Applying Itô's formula, for λ ≥ λ T we deduce from (41) that This, in addition to integration by parts, further implies that As a consequence, the desired assertion (40) is now available.
The following lemma concerns the continuity in the mean L 2 -norm sense for the displacement of (Y Proof. To make the content self-contained, we here give a sketch although the corresponding argument of (42) is quite standard. By virtue of (36), it follows immediately that Recall the elementary inequalities: for any η ∈ (0, 1], Next, according to Hölder's inequality and Itô's isometry and by taking contractive property of e tA and b T,∞ < ∞ into account, we derive from (43) that where in the penultimate display we used (20) with θ = α and in the last step utilized (3).